The excited-state quantum dynamics of the organic cation in hybrid perovskites are investigated using the time-dependent density functional theory. The bond fluctuation reveals that the energy relaxation follows different pathways depending on the chemical bonding characteristics within the cation molecule, which can fundamentally affect photostability. For the ammonium-group-containing cations, such as methylammonium (MA) or ethylammonium (EA), local vibrational modes survive for a long time. However, as their lowest unoccupied molecular orbitals (LUMOs) have pi* characters, the amidinium-group-containing cations, such as formamidinium (FA) or guanidinium (GA), efficiently dissipate deposited energy via chaotic intramolecular vibrational energy redistribution. The distinct A-site molecules' dynamics are closely related to the quantum ergodicity, which can bring enhanced photostability of FA and GA compared to MA and EA. Our theoretical investigation reveals the quantum chaos origin of better light stability of FA-based perovskites and serves as the future research direction of the A-site engineering for better solar cells and light-emitting devices.